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3.4 Marginal Functions in Economics Marginal Analysis • Marginal analysis is the study of the rate of change of economic quantities. – An economist is not merely concerned with the value of an economy's gross domestic product (GDP) at a given time but is equally concerned with the rate at which it is growing or declining. – A manufacturer is not only interested in the total cost of corresponding to a certain level of production of a commodity, but also is interested in the rate of change of the total cost with respect to the level of production. Supply • In a competitive market, a relationship exists between the unit price of a commodity and the commodity’s availability in the market. • In general, an increase in the commodity’s unit price induces the producer to increase the supply of the commodity. • The higher unit price, the more the producer is willing to produce. Supply Equation • The equation that expresses the relation between the unit price and the quantity supplied is called a supply equation defined by p f x . • In general, p f x increases as x increases. Demand • In a free-market economy, consumer demand for a particular commodity depends on the commodity’s unit price. • A demand equation p f x expresses the relationship between the unit price p and the quantity demanded x. • In general, p f x decreases as x increases. • The more you want to buy, the unit price should be less. Cost Functions • The total cost is the cost of operating a business. Usually includes fixed costs and variable costs. • The cost function C(x) is a function of the total cost of operating a business. • The actual cost incurred in producing an additional unit of a certain commodity given that a plant is already at a level of operation is called the marginal cost. Rate of Change of Cost Function Suppose the total cost in dollars incurred each week by Polaraire for manufacturing x refrigerators is given by the total cost function C x 8000 200 x 0.2 x 2 (0 x 400) a. What is the actual cost incurred for manufacturing the 251st refrigerator? b. Find the rate of change of the total cost function with respect to x when x 250 . a. the actual cost incurred for manufacturing the 251st refrigerator is C 251 C 250 8000 200250 0.2250 8000 200251 0.2251 2 2 45,599.8 45,500 99.80 b. The rate of change is given by the derivative C' x 200 0.4 x Thus, when the level of production is 250 refrigerators, the rate of change of the total cost is C' 250 200 0.4250 100 • Observe that we can rewrite C 251 C 250 C 251 C 250 1 C 250 1 C 250 C 250 h C 250 1 h • The definition of derivative tells us that C 250 h C 250 C ' 250 lim h 0 h • Thus, the derivative C' x is a good approximation of the average rate of change of the function C x . Marginal Cost Function • The marginal cost function is defined to be the derivative of the corresponding total cost function. • If C x is the cost function, then C' x is its marginal cost function. • The adjective marginal is synonymous with derivative of. Revenue Functions • A revenue function R(x) gives the revenue realized by a company from the sale of x units of a certain commodity. • If the company charges p dollars per unit, then Rx px . • The demand function p f x tells the relationship between p and x. Thus, Rx xf x Marginal Revenue Functions • The marginal revenue function gives the actual revenue realized from the sale of an additional unit of the commodity given that sales are already at a certain level. • We define the marginal revenue function to be R' x . Profit Functions • The profit function is given by Px Rx Cx where R and C are the revenue and cost functions and x is the number of units of a commodity produced and sold. • The marginal profit function P' x measures the rate of change of the profit function and provides us with a good approximation of the actual profit or loss realized from the sale of the additional unit of the commodity. Average Cost Function • The average cost of producing units of the commodity is obtained by dividing the total production cost by the number of units produced. • The average cost function is denoted by C x and defined by C x x • The marginal average cost function C' x measures the rate of change of the average cost. The weekly demand for the Pulser 25 color LED television is p 600 0.05x (0 x 12,000) where p denotes the wholesale unit price in dollars and x denotes the quantity demanded. The weekly total cost function associated with manufacturing the Pulser 25 is given by C x 0.000002 x 0.03x 400 x 80,000 3 2 where C(x) denotes the total cost incurred in producing x sets. a. Find the revenue function R and the profit function P. R x px 600 0.05 x x 600 x 0.05 x 2 P x R x C x 600 x 0.05 x 2 0.000002 x 0.03x 400 x 80,000 3 2 0.000002 x 0.02 x 200 x 80,000 3 2 b. Find the marginal cost function, the marginal revenue function, and the marginal profit function. C ' x 0.000006 x 0.06 x 400 2 R' x 600 0.1x P' x 0.000006 x 0.04 x 200 2 c. Compute C ' 2000 , R' 2000 , and P' 2000 and interpret your results. • • • Since C' 2000 304 , the cost to manufacture the 2001st LED TV is approximately 304. Since R' 2000 400 , the revenue increased by manufacturing the 2001st LED TV is approximately 400. Since P' 2000 96 , the profit for manufacturing and selling the 2001st LED TV is approximately 96. Elasticity of Demand • Question: when you produce more commodities, do you actually get the more revenue? • It is convenient to write the demand function f in the form x f p ; that is, we will think of the quantity demanded of a certain commodity as a function of its unit price. • Usually, when the unit price of a commodity increases, the quantity demanded decreases Suppose the unit price of a commodity is increased by h dollars from p dollars to p+ h dollars. Then the quantity demanded drops from f (p) units to f(p + h) units. The percentage change in the unit price is h 100 p and the corresponding percentage change in the quantity demanded is f p h f p 100 f p Percentage change in the quantity demanded Percentage change in the unit price f p h f p 100 f p h 100 p p f p h f p f p h Elasticity of Demand f p h f p f ' p h • Since previous ratio as , we can write the pf ' p E p f p called the elasticity of demand at price p. • We will see in section 4.1 that f ' p 0 since f is decreasing. Because economists would rather work with a positive value, we put a negative sign. Consider the demand equation p 0.02 x 400 (0 x 20,000) which describes the relationship between the unit price in dollars and the quantity demanded x of the Acrosonic model F loudspeaker systems. Find the elasticity of demand E(p). pf ' p p 50 E p f p 50 p 20,000 p 400 p • When p 100 1 , we have E 100 . This result 3 tells us that when the unit price is set at $100 per speaker, an increase of 1% in the unit price will cause a decrease of approximately 0.33% in the quantity demanded. • When p 300 , we have E 300 3. This result tells us that when the unit price is set at $100 per speaker, an increase of 1% in the unit price will cause a decrease of approximately 0.33% in the quantity demanded. Since the revenue is R p px pf p , the marginal revenue function is R' p f p pf ' p pf ' p f p 1 f p 1 E p f p 1 • Since E 100 1, we have R' 100 0 . If we 3 increase the unit price, the revenue increases. • Since E300 3 1 , we have R' 300 0. If we decrease the unit price, the revenue decreases. • If the demand is elastic at p [E(p)>1], then an increase in the unit price will cause the revenue to decrease, whereas a decrease in the unit price will cause the revenue to increase. • If the demand is inelastic at p [E(p)>1], then an increase in the unit price will cause the revenue to increase, whereas a decrease in the unit price will cause the revenue to decrease. • If the demand is unitary at p [E(p)>1], then an increase in the unit price will cause the revenue to stay about the same. Consider the demand equation p 0.01x 2 0.2 x 8 (0 x 20,000) and the quantity demanded each week is 15. If we increase the unit price a little bit, what will happen to our revenue? dx dx 1.2 1 0.02 x 0.2 dp dp 0.02 x When x=15, p=2.75, f(p)=x=15, and f ' p pf ' p 2.75 4 11 E p f p 15 15 dx 4 dp